What is PARALLEL AXIS THEOREM

Introduction

According toParallel axis theorem, the moment of inertia of any object through its center of mass is the minimum moment of inertia for an axis in that particular direction in the space. The moment of inertia about any axis parallel to that axis through the center of mass is given by

In the given expression, thecentre of massmoment of inertia will be recognized as the moment of inertia of a point mass - the moment of inertia about a parallel axis is the centre of mass moment plus the moment of inertia of the entire object treated as a point mass at the centre of mass.

Proof

The perpendicular distance between the axes lies along thex-axis and that the centre of mass lies at the origin. The moment of inertia relative to thez-axis, passing through the centre of mass, is:

The moment of inertia relative to the new axis, perpendicular distanceralong thex-axis from the centre of mass, is:

The first term isIcm, the second term becomesmr2, and the final term is zero since the origin is at the centre of mass. So, this expression becomes:

In classical mechanics

In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensorJijfrom an inertia tensor about a center of massIijwhen the pivot point is a displacementafrom the center of mass:

Jij=Iij+M(a2δij−aiaj)

where

is the displacement vector from the center of mass to the new axis, and

δij

is the Kronecker delta.

We can see that, for diagonal elements (wheni=j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.

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